The country in the world that comes closest to having a direct democracy is Switzerland. There is no country that has a truly direct democracy, but Switzerland comes closest. There are states in the US that have provisions for direct democracy as well.
Direct democracy occurs when all of the citizens of a polity participate in making decisions. A true direct democracy would have laws and decisions being made through meetings of all interested citizens. This is, of course, not feasible in any polity that has more than a few hundred people in it. There are a few small towns in New England that pass laws through town meetings, but no polity of any size can do this.
Instead, we have places like Switzerland that give the people the chance to approve or reject laws that have already been made. In Switzerland, any law that has been passed can be subjected to a public vote if enough people sign a petition asking for such a vote. In some US states, citizens who can gather enough signatures on their petitions can propose laws that are voted on by all the people of the state. These are examples of places that have aspects of direct democracy. There is no country that has a truly direct democracy.
An example of a place that follows direct democracy are the Swiss cantons (these are similar to states) Appenzell Innerrhoden and Glarus. In general Switzerland is a semi direct democracy as it is actually a representative democracy that contains elements of direct democracy. However in the case of these two cantons citizens have more power than they would in a tradition representative democracy as they can propose amendments.
Great question! Only small local governments are able to participate in direct democracy. So to find a place that follows direct or pure democracy today you would have to look at small town governments. However, the initiative and referendum processes allow citizens to place issues on a ballot at the State and Local level in many parts of the United States.
All States in the U.S. allow their representative bodies, i.e., their legislatures, to place issues on the ballot, but only 27 States have adopted one or both of two provisions that directly involves the citizens. Of those, 24 allow the initiative process and 23 allow both the initiative and referendum. States that allow one or the other are: Maryland and New Mexico with only the referendum; Florida, Illinois, and Mississippi with only the initiative. But that still leaves a long list of States who allow neither.
The Initiative and Referendum Institute website (use the link in the sources below) hosted by the University of California, provides a map that shows what States in the United States allow the initiative and referendum process. The website also provides the rules regarding how the two processes work in each state.
The earliest state to have adopted both of these measures is South Dakota which did so in 1898.
All states require popular vote to amend their constitutions EXCEPT Delaware.